Average Error: 34.1 → 10.0
Time: 17.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r96654 = b;
        double r96655 = -r96654;
        double r96656 = r96654 * r96654;
        double r96657 = 4.0;
        double r96658 = a;
        double r96659 = r96657 * r96658;
        double r96660 = c;
        double r96661 = r96659 * r96660;
        double r96662 = r96656 - r96661;
        double r96663 = sqrt(r96662);
        double r96664 = r96655 + r96663;
        double r96665 = 2.0;
        double r96666 = r96665 * r96658;
        double r96667 = r96664 / r96666;
        return r96667;
}
↓
double f(double a, double b, double c) {
        double r96668 = b;
        double r96669 = -1.119811545308531e+143;
        bool r96670 = r96668 <= r96669;
        double r96671 = 1.0;
        double r96672 = c;
        double r96673 = r96672 / r96668;
        double r96674 = a;
        double r96675 = r96668 / r96674;
        double r96676 = r96673 - r96675;
        double r96677 = r96671 * r96676;
        double r96678 = 4.718890261991469e-106;
        bool r96679 = r96668 <= r96678;
        double r96680 = r96668 * r96668;
        double r96681 = 4.0;
        double r96682 = r96681 * r96674;
        double r96683 = r96682 * r96672;
        double r96684 = r96680 - r96683;
        double r96685 = sqrt(r96684);
        double r96686 = r96685 - r96668;
        double r96687 = 2.0;
        double r96688 = r96687 * r96674;
        double r96689 = r96686 / r96688;
        double r96690 = -1.0;
        double r96691 = r96690 * r96673;
        double r96692 = r96679 ? r96689 : r96691;
        double r96693 = r96670 ? r96677 : r96692;
        return r96693;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.1
Target21.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.119811545308531e+143

    1. Initial program 59.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified59.0

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}}\]

    3. Taylor expanded around -inf 2.4

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    4. Simplified2.4

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -1.119811545308531e+143 < b < 4.718890261991469e-106

    1. Initial program 11.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified11.1

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}}\]
    3. Using strategy rm

    4. Applied div-inv11.2

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \frac{1}{2 \cdot a}}\]
    5. Using strategy rm

    6. Applied pow111.2

      \[\leadsto \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{1}}\]

    7. Applied pow111.2

      \[\leadsto \color{blue}{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}^{1}} \cdot {\left(\frac{1}{2 \cdot a}\right)}^{1}\]

    8. Applied pow-prod-down11.2

      \[\leadsto \color{blue}{{\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \frac{1}{2 \cdot a}\right)}^{1}}\]

    9. Simplified11.1

      \[\leadsto {\color{blue}{\left(\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\right)}}^{1}\]

    if 4.718890261991469e-106 < b

    1. Initial program 52.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified52.4

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}}\]

    3. Taylor expanded around inf 10.9

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))